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Brink, K. H., 2012. Baroclinic Instability of an Idealized Tidal Mixing Front

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Brink, K. H., 2012. Baroclinic Instability of an Idealized Tidal Mixing Front Journal of Marine Research, 70, 661–688, 2012 Baroclinic instability of an idealized tidal mixing front by K. H. Brink1 ABSTRACT Tidal mixing fronts separate vertically homogenized waters from stratified ambient waters. The linear and nonlinear baroclinic stability of an idealized tidal mixing front is treated here in the param- eter range that is stable with regard to symmetric instabilities and that has no bottom friction. All model configurations considered are unstable, and the dependence on bottom slope, stratification and other parameters is similar to that suggested by models (such as that of Blumsack and Gierasch, 1972) with much simpler configurations. The finite-amplitude evolution of the instabilities is treated using a primitive equation numerical model. The initial length scale and growth rate of the instabilities are well predicted by the linear calculations. As the system evolves, gravitational potential energy is trans- ferred to eddy kinetic energy, which peaks at about the time that potential energy stops decreasing. The peak eddy kinetic energy depends strongly on the bottom slope, with the greatest values occurring when the bottom and near-bottom isopycnals slope in the same direction. As the fields continue to evolve, eddy kinetic energy decreases, mean kinetic energy increases, and the eddies become larger and more barotropic. The horizontal eddy mixing coefficient is estimated at the time of maximum lateral heat flux and is found to be sensitive to the magnitude of the bottom slope but not its sign. Overall, the instability and the related eddy mixing are strong enough to encourage the idea that these instabilities might be effective at a more realistic tidal mixing front. 1. Introduction Among the most biologically productive regions in the global coastal ocean are those vertically homogenized by tidally induced turbulence, which are typically bounded by a tidal mixing front. This front (e.g., Fig. 1) separates shallower, well-mixed waters from deeper, stratified waters, and it occurs where the stabilizing surface heat fluxes exactly balance the tendency of tidally driven bottom boundary layer turbulence to homogenize the water column. The front’s location thus obeys the famous h/q3 criterion (where h is the water depth and q is a representative tidal current amplitude) (Simpson and Hunter, 1974), which requires that tidal mixing fronts generally parallel topography and thus usually intersect a sloping bottom. One substantial question arises about the productivity of these tidally mixed areas. Specif- ically, if biological activity in the mixed waters is sustained over time, one would expect a 1. Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, 02543, U.S.A. email: [email protected] 661 662 Journal of Marine Research [70, 4 Figure 1. A typical density (σt ) section across the northern side of Georges Bank, July 3, 1988. From Loder et al. (1992). An edited version of this paper was published by the American Geophysical Union. Copyright 1992 American Geophysical Union. flux of nutrients from the stratified ambient waters into the mixed region. This follows from the idea that the productivity cannot rely completely on recycled nutrients. Franks and Chen (1996) provide a very reasonable replication of this flux in a two-dimensional numerical model of tides and the mixing front. The two-dimensional model, however, does not embrace the only potential means to transport nutrients through a tidal mixing front. Garrett and Loder (1981), for example, suggest a range of possibilities that include wind driving and eddy transports associated with frontal instabilities. Since then, steady circulations have been investigated more deeply by Loder and Wright (1985), and wind driving by Chen et al. (2003). There is plentiful evidence for tidal mixing front instability around the British Isles (e.g., Simpson and James, 1986), although this possibility is rarely mentioned in the context of Georges Bank, a tidally mixed area off the eastern coast of North America. There have been a number of studies of the stability of tidal mixing fronts, but these have been very idealized or realistically complex. In the idealized category, van Heijst (1986) treats an inviscid case with a flat bottom and two immiscible layers. Some laboratory experiments (van Heijst, 1986; Thomas and Linden, 1996) also treat similarly idealized situations. Among numerical models, James (1989) treats a simple case but without tides, while Badin et al. (2009) find evidence for frontal instability in their very complete simulation runs. As an instability problem, the tidal mixing front has a couple of interesting, relatively unique features. One is that the origin of the front depends on waters being vertically 2012] Brink: Baroclinic instability of an idealized tidal mixing front 663 homogenized by turbulence generated at the bottom. Thus, in a realistic context, bottom stress is of lowest order importance in defining the basic state and, presumably, for a growing instability as well. Second, the front itself is expected to move back and forth across isobaths with the tides. This sort of oscillation in the background flow has been shown, for a simpler context, to enhance the growth of instabilities under at least some conditions (Poulin et al., 2003; Flierl and Pedlosky, 2007). Their purpose of this contribution is to provide a first step in a systematic study of instability and associated cross-frontal exchange at a tidal mixing front. The starting point is to consider steady fronts that have the structure of an idealized tidal mixing front but that are not affected by bottom friction. Further, attention here is also confined to the case where the front is stable with regard to symmetric instabilities (e.g., Stone, 1966). Given these constraints, the linear instability and finite-amplitude evolution of the front still require an investigation of a parameter space that includes the influences of bottom slope, stratification, frontal width and rotation. Future contributions will deal first with slantwise convection and bottom friction, and later with a realistic, oscillating, tidal mixing front. In the end, while frontal instability is central to the issues at hand, the real goal is to quantify the extent to which the resulting eddy field can effect exchanges across the front. One important point of comparison in the following calculations is the quasigeostrophic linear stability model of Blumsack and Gierasch (1972: BG72 henceforth), who considered a case with spatially uniform stratification and constant vertical shear over a uniformly sloping bottom. When the bottom is flat, the maximum growth rate occurs at a scale similar to the internal Rossby radius of deformation, and the instability has no long-wave cutoff. When the bottom slopes in the opposite direction from the isopycnal slope (positive slope, in the notation of the following sections), the maximum instability occurs at shorter scales, while the maximum growth rate decreases, but never vanishes, with increasing slope. When the bottom slopes in the opposite sense (i.e., in the same direction as isopycnals), the maximum growth rate and the most unstable wavelength both increase until they reach a maximum. Beyond this point, both the growth rate and wavelength decrease until the instability vanishes when the bottom slope exceeds the isopycnal slope. There is always a short wavelength cutoff for instability, and there is a long-wave cutoff, as well, when the bottom slope is nonzero. Overall, a bottom slope in the same direction as the isopycnal tilt increases instability (up to a point), and the opposite slope decreases the instability. At first glance, this might seem to contradict Barth’s (1989) findings for an upwelling front, but in his case, the tilting isopycnals intersect a level upper surface, and not the sloping bottom. Despite the very simple (as compared with a tidal mixing front) configuration of BG72, their results provide useful context in the following. 2. Formulation a. The physical configuration In all cases treated here, the basic state (Fig. 2) consists of mean flow along the axis of a channel of width W. One side of the channel (larger x)has uniform density stratification, and 664 Journal of Marine Research [70, 4 Figure 2. Representative initial conditions for either a linear or nonlinear stability calculation, here for case 4 (Table 1). The left panel is initial density (contour interval: 0.1 σt unit). The right panel is initial along-channel velocity. Positive and zero contours are heavy, and negative contours are lighter lines (contour interval: 2 cm s−1). buoyancy frequency N0. The opposite side of the channel has very weak initial stratification, if any. Between these two regions, there is a mean along-channel flow v¯ in thermal wind balance with the mean density field ρ¯. The level of no motion is at the mean depth of the channel, H . Specifically, the fields are given by ρ¯(x, z) = ρ0 for x<W/2 − L, (1a) π(x − W/2) ρ¯(x, z) = ρ 1 − 1 + sin N 2(z − z )/(2g) 0 2L 0 0 for W/2 − L<x<W/2 + L, and (1b) ρ¯ = ρ [ − 2 − ] + (x, z) 0 1 N0 (z z0)/g for x>W/2 L, (1c) so that v¯(x, z) = 0 for |x − W/2| >L,and (2a) π π(x − W/2) v(x, z) = cos N 2(z + H)[(z − H)− 2z ] 8fL 2L 0 0 for |x − W/2| <L. (2b) 2012] Brink: Baroclinic instability of an idealized tidal mixing front 665 The Coriolis parameter is f, the acceleration due to gravity is g, ρ0 is the background density of the water, −z0 is the depth of the velocity maximum and L is the half width of the frontal region. Attention here is restricted to the range where the maximum velocity occurs at or above mid-depth: i.e., where −H/2 ≤ z0 ≤ 0, so that the mean along-channel flow is in the same direction for −H ≤ z.
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